A strong loophole-free test of localrealism

Shalm, Meyer-Scott, Christensen, Bierhorst, Wayne, Stevens, Gerrits,Glancy, Hamel, Allman, Coakley, Dyer, Hodge, Lita, Verma, Lambrocco,Tortorici, Migdall, Zhang, Kumor, Farr, Marsili, Shaw, Stern, Abellán,

Amaya, Pruneri, Jennewein, Mitchell, Kwiat, Bienfang, Mirin, Knill, and Nam

Nov 10, 2015

Abstract

We present a loophole-free violation of local realism using entangled pho-ton pairs. We ensure that all relevant events in our Bell test are space-like separated by placing the parties far enough apart and by using fastrandom number generators and high-speed polarization measurements. Ahigh-quality polarization-entangled source of photons, combined with high-efficiency, low-noise, single-photon detectors, allows us to make measure-ments without requiring any fair-sampling assumptions. Using a hypothesistest, we compute p-values as small as 5.9 × 10−9 for our Bell violation whilemaintaining the spacelike separation of our events. We estimate the degreeto which a local realistic system could predict our measurement choices. Ac-counting for this predictability, our smallest adjusted p-value is 2.3 × 10−7.We therefore reject the hypothesis that local realism governs our experiment.

But if [a hidden variable theory] is local it will not agree with quantum me-chanics, and if it agrees with quantum mechanics it will not be local. This iswhat the theorem says. - John Stewart Bell

Quantum mechanics at its heart is a statistical theory. It cannot with cer-tainty predict the outcome of all single events, but instead it predicts proba-bilities of outcomes. This probabilistic nature of quantum theory is at oddswith the determinism inherent in Newtonian physics and relativity, whereoutcomes can be exactly predicted given sufficient knowledge of a system.Einstein and others felt that quantum mechanics was incomplete. Perhapsquantum systems are controlled by variables, possibly hidden from us [2],that determine the outcomes of measurements. If we had direct access tothese hidden variables then the properties of quantum systems would notneed to be treated probabilistically. De Broglie’s 1927 pilot-wave theory wasa first attempt at formulating a hidden variable theory of quantum physics[3]; it was completed in 1952 by David Bohm [4, 5]. While the pilot-wavetheory can reproduce all of the predictions of quantum mechanics, it has thecurious feature that hidden variables in one location can instantly change val-ues because of events happening in distant locations. This seemingly violatesthe locality principle from relativity, which says that objects cannot signalone another faster than the speed of light. In 1935 the nonlocal feature ofquantum systems was popularized by Einstein, Podolsky, and Rosen [6], andis something Einstein later referred to as “spooky actions at a distance”[7].

But in 1964 John Bell showed that it is impossible to construct a hiddenvariable theory that obeys locality and simultaneously reproduces all of thepredictions of quantum mechanics [8]. Bell’s theorem fundamentally changedour understanding of quantum theory and today stands as a cornerstone ofmodern quantum information science.

Bell’s theorem does not prove the validity of quantum mechanics, but it doesallows us to test the hypothesis that nature is governed by local realism.The principle of realism says that any system has pre-existing values for allpossible measurements of the system. In local realistic theories, these pre-existing values depend only on events in the past lightcone of the system.Local hidden-variable theories obey this principle of local realism. Localrealism places constraints on the behavior of systems of multiple particles—constraints that do not apply to entangled quantum particles. This leads todifferent predictions that can be tested in an experiment known as a Bell test.In a typical two-party Bell test, a source generates particles and sends themto two distant parties, Alice and Bob. Alice and Bob independently andrandomly choose properties of their individual particles to measure. Later,they compare the results of their measurements. Local realism constrains thejoint probability distribution of their choices and measurements. The basisof a Bell test is an inequality that is obeyed by local realistic probabilitydistributions but can be violated by the probability distributions of certainentangled quantum particles [8]. A few years after Bell derived his inequality,new forms were introduced by Clauser, Horne, Shimony and Holt [9], andClauser and Horne [10] that are simpler to experimentally test.

In a series of landmark experiments, Freedman and Clauser [11] and Aspect,Grangier, Dalibard, and Roger [12-14] demonstrated experimental violationsof Bell inequalities using pairs of polarization-entangled photons generatedby an atomic cascade. However, due to technological constraints, these Belltests and those that followed (see [15] for a review) were forced to makeadditional assumptions to show local realism was incompatible with theirexperimental results. A significant violation of Bell’s inequality implies thateither local realism is false or that one or more of the assumptions madeabout the experiment are not true; thus every assumption in an experimentopens a “loophole”. No experiment can be absolutely free of all loopholes,but in [16] a minimal set of assumptions is described that an experimentmust make to be considered “loophole free”. Here we report a significant,

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loophole free, experimental violation of local realism using entangled photonpairs. We use the definition of loophole free as defined in [16]. In our ex-periment the only assumptions that remain are those that can never—evenin principle—be removed. We present physical arguments and evidence thatthese remaining assumptions are either true or untestable.

Bell’s proof requires that the measurement choice at Alice cannot influencethe outcome at Bob (and vice-versa). If a signal traveling from Alice can notreach Bob in the time between Alice’s choice and the completion of Bob’smeasurement, then there is no way for a local hidden variable constrainedby special relativity at Alice to change Bob’s outcomes. In this case we saythat Alice and Bob are spacelike separated from one another. If an exper-iment does not have this spacelike separation, then an assumption must bemade that local hidden variables cannot signal one another, leading to the“locality” loophole.

Another requirement in a Bell test is that Alice and Bob must be free tomake random measurement choices that are physically independent of oneanother and of any properties of the particles. If this is not true, then ahidden variable could predict the chosen settings in advance and use thatinformation to produce measurement outcomes that violate a Bell inequal-ity. Not fulfilling this requirement opens the “freedom-of-choice” loophole.While this loophole can never in principle be closed, the set of hidden vari-able models that are able to predict the choices can be constrained usingspace-like separation. In particular, in experiments that use processes suchas cascade emission or parametric down-conversion to create entangled parti-cles, space-like separation of the measurement choices from the creation eventeliminates the possibility that the particles, or any other signal emanatingfrom the creation event, influence the settings. To satisfy this condition, Al-ice and Bob must choose measurement settings based on fast random eventsthat occur in the short time before a signal traveling at the speed of light fromthe entangled-photon creation would be able to reach them. But it is fun-damentally impossible to conclusively prove that Alice’s and Bob’s randomnumber generators are independent without making additional assumptions,since their backward lightcones necessarily intersect. Instead, it is possibleto justify the assumption of measurement independence through a detailedcharacterization of the physical properties of the random number generators(such as the examination described in [17, 18]).

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In any experiment, imperfections could lead to loss, and not all particles willbe detected. To violate a Bell inequality in an experiment with two parties,each free to choose between two settings, Eberhard showed that at least 2/3of the particles must be detected [19] if non-maximally entangled states areused. If the loss exceeds this threshold, then one may observe a violation bydiscarding events in which at least one party does not detect a particle. Thisis valid under the assumption that particles were lost in an unbiased manner.However, relying on this assumption opens the “detector” or “fair-sampling”loophole. While the locality and fair-sampling loopholes have been closedindividually in different systems [20-24], it has only recently been possiblein our system and in two others [25, 26] to close all loopholes simultane-ously. These three experiments also address the freedom-of-choice loopholeby space-like separation.

Fundamentally a Bell inequality is a constraint on probabilities that are es-timated from random data. Determining whether a data set shows violationis a statistical hypothesis-testing problem. It is critical that the statisticalanalysis does not introduce unnecessary assumptions that create loopholes.A Bell test is divided into a series of trials. In our experiment, during eachtrial Alice and Bob randomly choose between one of two measurement set-tings (denoted {a, a′} for Alice and {b, b′} for Bob) and record either a “+”if they observe any detection events or a “0” otherwise. Alice and Bob mustdefine when a trial is happening using only locally available information,otherwise additional loopholes are introduced. At the end of the experimentAlice and Bob compare the results they obtained on a trial-by-trial basis.Our Bell test uses a version of the Clauser-Horne/Eberhard inequality [10,19, 27] where, according to local realism,

P (+ + ∣ab) ≤ P (+0∣ab′) + P (0 + ∣a′b) + P (+ + ∣a′b′) (1)

The terms P (+ + ∣ab) and P (+ + ∣a′b′) correspond to the probability thatboth Alice and Bob record detection events (++) when they choose the mea-surement settings ab or a′b′, respectively. Similarly, the terms P (+0∣ab′) andP (0 + ∣a′b) are the probabilities that only Alice or Bob record an event forsettings ab′ and a′b, respectively. A local realistic model can saturate thisinequality; how- ever, the probability distributions of entangled quantumparticles can violate it.

To quantify our Bell violation we construct a hypothesis test based on the

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inequality in Eq. (1). The null hypothesis we test is that the measuredprobability distributions in our experiment are constrained by local realism.Our evidence against this null hypothesis of local realism is quantified in ap-value that we compute from our measured data using a test statistic. Ourtest statistic takes all of the measured data from Alice’s and Bob’s trialsand summarizes them into a single number (see the Supplemental Materialfor further details). The p-value is then the maximum probability that ourexperiment, if it is governed by local realism, could have produced a value ofthe test statistic that is at least as large as the observed value [28]. Smaller p-values can be interpreted as stronger evidence against this hypothesis. Thesep-values can also be used as certificates for cryptographic applications, suchas random number generation, that rely on a Bell test [24, 29]. We use amartingale binomial technique [27] for computing the p-value that makes noassumptions about the distribution of events and does not require that thedata be independent and identically distributed [30] as long as appropriatestopping criteria are determined in advance.

In our experiment, the source creates polarization- entangled pairs of pho-tons and distributes them to Alice and Bob, located in distant labs. At thesource location a mode-locked Ti:Sapphire laser running at repetition rateof approximately 79.3 MHz produces picosecond pulses centered at a wave-length of 775 nm as shown in figure 1. These laser pulses pump an apodizedperiodically poled potassium titanyl phosphate (PPKTP) crystal to producephoton pairs at a wavelength of 1550nm via the process of spontaneous para-metric downconversion [31]. The downconversion system was designed usingthe tools available in [32]. The PPKTP crystal is embedded in the middle ofa polarization-based Mach-Zehnder interferometer that enables high-qualitypolarization-entangled states to be generated [33]. Rotating the polarizationanalyzer angles at Alice and Bob, we measure the visibility of coincidencedetections for a maximally entangled state to be 0.999±0.001 in the horizon-tal/vertical polarization basis and 0.996± 0.001 in the diagonal/antidiagonalpolarization basis (see [34] for information about the reported uncertainties).The entangled photons are then coupled into separate single-mode opticalfibers with one photon sent to Alice and the other to Bob. Alice, Bob, andthe source are positioned at the vertices of a nearly right-angle triangle.

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Figure 1: Schematic of the entangled photon source. A pulsed 775 nm-wavelengthTi:Sapphire picosecond mode-locked laser running at 79.3 MHz repetition rate is usedas both a clock and a pump in our setup. A fast photodiode (FPD) and divider circuitare used to generate the synchronization signal that is distributed to Alice and Bob.A polarization-maintaining single-mode fiber (SMF) then acts as a spatial filter for thepump. After exiting the SMF, a polarizer and half-wave plate (HWP) set the pumppolarization. To generate entanglement, a periodically poled potassium titanyl phosphate(PPKTP) crystal designed for Type-II phasematching is placed in a polarization-basedMach-Zehnder interferometer formed using a series of HWPs and three beam displacers(BD). At BD1 the pump beam is split in two paths (1 and 2): the horizontal (H) componentof polarization of the pump translates laterally in the x direction while the vertical (V)component of polarization passes straight through. Tilting BD1 sets the phase, φ, of theinterferometer to 0. After BD1 the pump state is (cos (16○) ∣H1⟩+sin (16

○) ∣V2⟩. To address

the polarization of the paths individually, semi-circular waveplates are used. A HWP inpath 2 rotates the polarization of the pump from vertical (V) to horizontal (H). A secondHWP at 0○ is inserted into path 1 to keep the path lengths of the interferometer balanced.The pump is focused at two spots in the crystal, and photon pairs at a wavelength of1550nm are generated in either path 1 or 2 through the process of spontaneous parametricdownconversion. After the crystal, BD2 walks the V-polarized signal photons down inthe y direction (V1a and V2a) while the H-polarized idler photons pass straight through(H1b and H2b). The x − y view shows the resulting locations of the four beam paths.HWPs at 45○ correct the polarization while HWPs at 0○ provide temporal compensation.BD3 then completes the interferometer by recombining paths 1 and 2 for the signal andidler photons. The two downconversion processes interfere with one another, creating theentangled state in Eq. (2). A high-purity silicon wafer with an anti-reflection coating isused to filter out the remaining pump light. The idler (signal) photons are coupled into aSMF and sent to Alice (Bob).

Due to constraints in the building layout, the photons travel to Alice and Bobin fiber optic cables that are not positioned along their direct lines of sight.While the photons are in flight toward Alice and Bob, their random num-

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ber generators each choose a measurement setting. Each choice is completedbefore information about the entangled state, generated at the PPKTP crys-tal, could possibly reach the random number generators. When the photonsarrive at Alice and Bob, they are launched into free space, and each pho-ton passes through a Pockels cell and polarizer that perform the polarizationmeasurement chosen by the random number generators as shown in Fig. 2.

Figure 2: Receiver station setup for Alice and Bob. A photon arrives from the source.Two half-wave plates (HWP), a quarter-wave plate (QWP), a Pockels cell (PC), and twoplate polarizers together act to measure the polarization state of the incoming photon.The polarization projection is determined by a random bit from XORing the outputs oftwo random number generators (RNG1 and RNG2) with pre-determined pseudorandombits (RNG3). If the random bit is “0”, corresponding to measurement setting a(b) for Alice(Bob), the Pockels cell remains off. If the random bit is “1”, corresponding to measurementsetting a?(b?) for Alice (Bob), then a voltage is applied to the Pockels cell that rotates thepolarization of the photons using a fast electro-optic effect. The two plate polarizers havea combined contrast ratio > 7000:1. The photons are coupled back into a single-mode fiber(SMF) and detected using a superconducting nanowire single-photon detector (SNSPD).The signal is amplified and sent to a time-tagging unit where the arrival time of the eventis recorded. The time tagger also records the measurement setting, the synchronizationsignal, and a one pulse-per-second signal from a global positioning system (GPS). Thepulse-per-second signal provides an external time reference that helps align the time tagsAlice and Bob record. A 10 MHz oscillator synchronizes the internal clocks on Alice’sand Bob’s time taggers. The synchronization pulse from the source is used to trigger themeasurement basis choice.

After the polarizer, the photons are coupled back into a single-mode fiberand sent to superconducting nanowire single-photon detectors, each with adetection efficiency of 91 ± 2% [35]. The detector signal is then amplifiedand sent to a time tagger where the arrival time is recorded. We assume the

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measurement outcome is fixed when it is recorded by the time tagger, whichhappens before information about the other party’s setting choice could pos-sibly arrive, as shown in Fig. 3(b).

Figure 3: Minkowski diagrams for the spacetime events related to Alice (A) and thesource (S) and Bob (B) and the source (a), and Alice and Bob (b). All lightcones areshaded blue. Due to the geometry of Alice, Bob, and the source, more than one spacetimediagram is required. In a) the random number generators (RNGs) at Alice and Bob mustfinish picking a setting outside the lightcone of the birth of an entangled photon pair. Atotal of 15 pump pulses have a chance of downconverting into an entangled pair of photonseach time the Pockels cells are on. The events related to the first pulse are not spacelikeseparated, because Alice’s RNG does not finish picking a setting before information aboutthe properties of the photon pair can arrive; pulses 2 through 11 are spacelike separated.As shown in (b), pulses 12 through 15 are not spacelike separated as the measurement isfinished by Alice and Bob after information about the other party’s measurement settingcould have arrived. In our experiment the events related to pulse 6 are the furthest outsideof all relevant lightcones.

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Alice and Bob have system detection efficiencies of 74.7±0.3% and 75.6±0.3%,respectively. We measure this system efficiency using the method outlined byKlyshko [36]. Background counts from blackbody radiation and room lightsreduce our observed violation of the Bell inequality. Every time a back-ground count is observed it counts as a detection event for only one party.These background counts increase the heralding efficiency required to closethe detector loophole above 2/3 [19]. To reduce the number of backgroundcounts, the only detection events considered are those that occur within awindow of approximately 625ps at Alice and 781 ps at Bob, centered aroundthe expected arrival times of photons from the source. The probability ofobserving a background count during a single window is 8.9 × 10−7 for Al-ice and 3.2 × 10−7 for Bob, while the probability that a single pump pulsedownconverts into a photon pair is ≈ 5 × 10−4. These background counts inour system raise the efficiency needed to violate a Bell inequality from 2/3to 72.5%. Given our system detection efficiencies, our entangled photon pro-duction rates, entanglement visibility, and the number of background counts,we numerically determine the entangled state and measurement settings forAlice and Bob that should give the largest Bell violation for our setup. Theoptimal state is not maximally entangled [19] and is given by:

∣ψ⟩ = 0.961 ∣HAHB⟩ + 0.276 ∣VAVB⟩ (2)

where H(V ) denotes horizontal (vertical) polarization, and A and B cor-respond to Alice’s and Bob’s photons, respectively. From the simulationwe also determine that Alice’s optimal polarization measurement angles,relative to a vertical polarizer, are {a = 4.2○, a′ = −25.9○} while Bob’s are{b = −4.2○, b′ = 25.9○}.

Synchronization signals enable Alice and Bob to define trials based only onlocal information. The synchronization signal runs at a frequency of 99.1kHz, allowing Alice and Bob to perform 99,100 trials/s (79.3 MHz/800).This trial frequency is limited by the rate the Pockels cells can be stablydriven. When the Pockels cells are triggered they stay on for ≈ 200 ns. Thisis more than 15 times longer than the 12.6ns pulse-to-pulse separation ofthe pump laser. Therefore photons generated by the source can arrive inone of 15 slots while both Alice’s and Bob’s Pockels cells are on. Since themajority of the photon pulses arriving in these 15 slots satisfy the spacelikeseparation constraints, it is possible to aggregate multiple adjacent pulsesto increase the event rate and statistical significance of the Bell violation.

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However, including too many pulses will cause one or more of the spacelikeseparation constraints to be violated. Because the probability per pulse ofgenerating an entangled photon pair is so low, given that one photon hasalready arrived, the chance of getting a second event in the same Pockels cellwindow is negligible (< 1%).

Alice and Bob each have three different sources of random bits that they XORtogether to produce their random measurement decisions (for more informa-tion see the Supplemental Materials). The first source is based on measuringoptical phase diffusion in a gain-switched laser that is driven above and be-low the lasing threshold. A new bit is produced every 5 ns by comparingadjacent laser pulses [17]. Each bit is then XORed with all past bits thathave been produced (for more details see the Supplemental Material). Thesecond source is based on sampling the amplitude of an optical pulse at thesingle-photon level in a short temporal interval. This source produces a biton demand and is triggered by the synchronization signal. Finally, Alice andBob each have a different predetermined pseudorandom source that is com-posed of various popular culture movies and TV shows, as well as the digitsof π, XORed together. Suppose that a local-realistic system with the goal ofproducing violation of the Bell inequality, was able to manipulate the prop-erties of the photons emitted by the entanglement source before each trial.Provided that the randomness sources correctly extract their bits from theunderlying processes of phase diffusion, optical amplitude sampling, and theproduction of cultural artifacts (such as the movie Back to the Future), thispowerful local realistic system would be required to predict the outcomes ofall of these processes well in advance of the beginning of each trial to achieveits goal. Such a model would have elements of superdeterminism—the fun-damentally untestable idea that all events in the universe are preordained.

Over the course of two days we took a total of 6 data runs with differing con-figurations of the experimental setup [37]. Here we report the results fromthe final dataset that recorded data for 30 minutes (see the SupplementalMaterial for descriptions and results from all datasets). This is the datasetwhere the experiment was most stable and best aligned; small changes in cou-pling efficiency and the stability of the Pockels cells can lead to large changesin the observed violation. The events corresponding to the sixth pulse outof the 15 possible pulses per trial are the farthest outside all the relevantlight-cones. Thus we say these events are the most spacelike separated. To

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increase our data rate we aggregate multiple pulses centered around pulsenumber 6. We consider different Bell tests using a single pulse (number 6),three pulses (pulses 5, 6, and 7), five pulses (pulses 4 through 8), and sevenpulses (pulses 3 through 9). The joint measurement outcomes and corre-sponding p-values for these combinations are shown in Table 1.

Aggregate N(+ + ∣ab) Nstop Total trials P-value Adjusted Timing MinimumPulses p-value Margin (ns) distance (m)

1 1257 2376 175,654,992 2.5 × 10−3 5.9 × 10−3 63.5 ± 3.7 9.23 3800 7211 175,744,824 2.4 × 10−6 2.3 × 10−5 50.9 ± 3.7 7.35 6378 12127 177,358,351 5.9 × 10−9 2.3 × 10−7 38.3 ± 3.7 5.47 8820 16979 177,797,650 2.0 × 10−7 9.2 × 10−6 25.7 ± 3.7 3.5

Table 1: P-value results for different numbers of aggregate pulses. Here N(+ + ∣ab)refers to the number of times Alice and Bob both detect a photon with settings a andb respectively. Before analyzing the data a stopping criteria, Nstop, was chosen. Thisstopping criteria refers to the total number of events considered that have the settings andoutcomes specified by the terms in Eq. (1), Nstop = N(+ + ∣ab) +N(+0∣ab

′) +N(0 + ∣a′b) +

N(+ + ∣a′b′). After this number of trials the p-value is computed and the remaining trialsdiscarded. Such pre-determined stopping criteria are necessary for the hypothesis test weuse (see Supplemental Material for more details). The total trials include all trials upto the stopping criteria regardless of whether a photon is detected. The adjusted p-valueaccounts for the excess predictability we estimate from measurements of one of our randomnumber generators. As discussed in the text, the time difference between Bob finishinghis measurement and the earliest time at which information about Alice’s measurementchoice could arrive at Bob sets the margin of timing error that can be tolerated and stillhave all events guaranteed to be spacelike separated. We also give the minimum distancebetween each party and its boundary line (shown in Fig. 4(a)) that guarantees satisfactionof the spacelike separation constraints. In the Supplemental Material the frequencies ofeach combination of settings choice for 5 aggregate pulses is reported.

For a single pulse we measure a p-value = 2.5×10−3, for three pulses a p-value= 2.4×10−6, for five pulses a p-value = 5.9×10−9 and for seven pulses a p-value= 2.0 × 10−7, corresponding to a strong violation of local realism.

If, trial-by-trial, a conspiratorial hidden variable (or attacker in cryptographicscenarios) has some measure of control over or knowledge about the settingschoices at Alice and Bob, then they could manipulate the outcomes to ob-serve a violation of a Bell inequality. Even if we weaken our assumption thatAlice’s and Bob’s setting choices are physically independent from the source,we can still compute valid p-values against the hypothesis of local realism.We characterize the lack of physical independence with predictability of our

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random number generators.

Figure 4: (a) The positions of Alice (A), Bob (B), and the source (S) in the buildingwhere the experiment was carried out. The insets show a magnified (×2) view of Alice’sand Bob’s locations. The white dots are the location of the random number generators(RNGs). The larger circle at each location has a radius of 1m and corresponds to ouruncertainty in the spatial position measurements. Alice, Bob, and the source can belocated anywhere within the green shaded regions and still have their events be spacelikeseparated. Boundaries are plotted for aggregates of one, three, five, and seven pulses.Each boundary is computed by keeping the chronology of events fixed, but allowing thedistance between the three parties to vary independently. In (b) the p-value of each of theindividual 15 pulses is shown. Overlayed on the plot are the aggregate pulse combinationsused in the contours in (a). The statistical significance of our Bell violation does notappear to depend on the spacelike separation of events. For reference and comparisonpurposes only, the corresponding number of standard deviations for a given p-value (for aone-sided normal distribution) are shown.

The “predictability”, P , of a random number generator is the probability

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with which an adversary or local realistic system could guess a given settingchoice. We use the parameter ε, the “excess predictability” to place an upperbound on the actual predictability of our random number generators:

P ≤ 1

2(1 + ε) (3)

In principle, it is impossible to measure predictability through statisticaltests of the random numbers, because they can be made to appear random,unbiased, and independent even if the excess predictability during each trialis nonzero. Extreme examples that could cause nonzero excess predictabilityinclude superdeterminism or a powerful and devious adversary with accessto the devices, but subtle technical issues can never be entirely ruled out.Greater levels of excess predictability lead to lower statistical confidence ina rejection of local realism. In Fig. 5 we show how different levels of ex-cess predictability change the statistical significance of our results [38] (seeSupplemental Materials for more details).

Figure 5: The p-value for different numbers of aggregate pulses as a function of the excesspredictability, ε, in Alice’s and Bob’s measurement settings. Larger levels of predictabilitycorrespond to a weakening of the assumption that the settings choices are physicallyindependent of the photon properties Alice and Bob measure. As in Fig. 4(b), the p-value equivalent confidence levels corresponding to the number of standard deviations ofa one-sided normal distribution are shown for reference.

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We can make estimates of the excess predictability in our system. From addi-tional measurements, we observe a bias of (1.08± 0.07)× 10−4 in the settingsreaching the XOR from the laser diffusion random source, which includessynchronization electronics as well as the random number generator. If thisbias is the only source of predictability in our system, this level of bias wouldcorrespond to an excess predictability of approximately 2 × 10−4. To be con-servative we use an excess predictability bound that is fifteen times larger,εp = 3 × 10−3 (see Supplemental Material for more details). If our experi-ment had excess predictability equal to εp our p-values would be increasedto 5.9 × 10−3, 2.4 × 10−5, 2.3 × 10−7, and 9.2 × 10−6 for one, three, five, andseven pulses, respectively [38]. Combining the output of this random numbergenerator with the others should lead to lower bias levels and a lower excesspredictability, but even under the the paranoid situation where a nearly su-perdeterministic local realistic system has complete knowledge of the bitsfrom the other random number sources, the adjusted p-values still provide arejection of local realism with high statistical significance.

Satisfying the spacetime separations constraints in Fig. 3 requires precisemeasurements of the locations of Alice, Bob, and the source as well as thetiming of all events. Using a combination of position measurements from aglobal positioning system (GPS) receiver and site surveying, we determinethe locations of Alice, Bob, and the source with an uncertainty of < 1m. Thisuncertainty is set by the physical size of the cryostat used to house our detec-tors and the uncertainty in the GPS coordinates. There are four events thatmust be spacelike separated: Alice’s and Bob’s measurement choice must befixed before any signal emanating from the photon creation event could arriveat their locations, and Alice and Bob must finish their measurements beforeinformation from the other party’s measurement choice could reach them.Due to the slight asymmetry in the locations of Alice, Bob, and the source,the time difference between Bob finishing his measurement and informationpossibly arriving about Alice’s measurement choice is al- ways shorter thanthe time differences of the other three events as shown in Fig. 3(b). Thistime difference serves as a kind of margin; our system can tolerate timingerrors as large as this margin and still have all events remain spacelike sepa-rated. For one, three, five, and seven aggregate pulses this corresponds to amargin of 63.5 ± 3.7 ns, 50.9 ± 3.7 ns, 38.3 ± 3.7 ns, and 25.7 ± 3.7 ns, respec-tively as shown in Table I. The uncertainty in these timing measurements isdominated by the 1 m positional uncertainty (see Supplemental Material for

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further details on the timing measurements).

A way to visualize and further quantify the the space- like separation ofevents is to compute how far Alice, Bob, and source could move from theirmeasured position and still be guaranteed to satisfy the locality constraints,assuming that the chronology of all events remains fixed. In figure 4(a) Alice,Bob, and the source locations are surrounded by shaded green regions. Aslong as each party remains anywhere inside the boundaries of these regionstheir events are guaranteed to be spacelike separated. There are specific con-figurations where all three parties can be outside the boundaries and stillbe spacelike separated, but here we consider the most conspiratorial casewhere all parties can collude with one another. The boundaries are over-layed on architectural drawings of the building in which the experiment wasperformed. Four different boundaries are plotted, corresponding to the Belltest performed with one, three, five, and seven aggregate pulses. Minimizingover the path of each boundary line, the minimum distance that Alice, Bob,and the source are located from their respective boundaries is 9.2 m, 7.3 m,5.4 m, and 3.5 m for aggregates of one pulse, three pulses, five pulses, andseven pulses, respectively. For these pulse configurations we would have hadto place our source and detection systems physically in different rooms (oreven move outside of the building) to compromise our spacelike separation.Aggregating more than seven pulses leads to boundaries that are less thanthree meters away from our measured positions. In these cases we are notable to make strong claims about the spacelike separation of our events.

Finally, as shown in Fig. 4(b), we can compute the 15 p-values for each of thetime slots we consider that pho- tons from the source can arrive in every trial.Photons arriving in slots 2 through 11 are spacelike separated while photonsin slots 12 through 15 are not. The photons arriving in these later slots aremeasured after information from the other party’s random number generatorcould arrive as shown in Fig. 3(b). It appears that spacelike separation hasno discernible effect on the statistical significance of the violation. However,we do see large slot-to-slot fluctuation in the calculated p-values. We sus-pect that this is due to instability in the applied voltage when the Pockelscell is turned on. In this case photons receive slightly different polarizationrotations depending on which slot they arrive in, leading to non-ideal mea-surement settings at Alice and Bob. It is because of this slot-to-slot variationthat the aggregate of seven pulses has a computed p-value larger than the

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five-pulse case. Fixing this instability and using more sophisticated hypoth-esis test techniques [39-41] will enable us to robustly increase the statisticalsignificance of our violation for the seven pulse case.

The experiment reported here is a commissioning run of the Bell test ma-chine we eventually plan to use to certify randomness. The ability to includemultiple pulses in our Bell test highlights the flexibility of our system. OurBell test machine is capable of high event rates, making it well suited forgenerating random numbers required by cryptographic applications [29]. Fu-ture work will focus on incorporating our Bell test machine as an additionalsource of real-time randomness into the National Institute of Standards andTechnology’s public random number beacon (https://beacon.nist.gov).

It has been 51 years since John Bell formulated his test of local realism.In that time his inequality has shaped our understanding of entanglementand quantum correlations, led to the quantum information revolution, andtransformed the study of quantum foundations. Until recently it has notbeen possible to carry out a complete and statistically significant loophole-free Bell test. Using advances in random number generation, photon sourcedevelopment, and high-efficiency single-photon detectors, we are able to ob-serve a strong violation of a Bell inequality that is loophole free, meaningthat we only need to make a minimal set of assumptions. These assumptionsare that our measurements of locations and times of events are reliable, thatAlice’s and Bob’s measurement outcomes are fixed at the time taggers, andthat during any given trial the random number generators at Alice and Bobare physically independent of each other and the properties of the photonsbeing measured. It is impossible, even in principle, to eliminate a form ofthese assumptions in any Bell test. Under these assumptions, if a hiddenvariable theory is local it does not agree with our results, and if it agreeswith our results then it is not local.

We thank Todd Harvey for assistance with optical fiber installation, KevinSilverman, Aephraim M. Stein- berg, Rupert Ursin, and Marissa Giustina forhelpful discussions, Nik Luhrs and Kristina Meier for helping with the elec-tronics, Andrew Novick for help with the GPS measurements, Joseph Chap-man and Malhar Jere for designing the cultural pseudorandom numbers, andStephen Jordan, Paul Lett, and Dietrich Leibfried for constructive commentson the manuscript. We thank Conrad Turner Bierhorst for waiting patiently

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for the computation of p-values. We dedicate this paper to the memory ofour coauthor, colleague, and friend, Jeffrey Stern. We acknowledge supportfor this project provided by: DARPA (LKS, MSA, AEL, SDD, MJS, VBV,TG, RPM, SWN, WHF, FM, MDS, JAS) and the NIST Quantum Informa-tion Program (LKS, MSA, AEL, SDD, MJS, VBV, TG, SG, PB, JCB, AM,RPM, EK, SWN); NSF grant No. PHY 12-05870 and MURI Center for Pho-tonic Quantum Information Systems (ARO/ARDA Program DAAD19-03-1-0199) DARPA InPho program and the Office of Naval Research MURI onFundamental Research on Wavelength-Agile High-Rate Quantum Key Dis-tribution (QKD) in a Marine Environment, award #N00014-13-0627 (BGC,MAW, DRK, PGK); NSERC, CIFAR and Industry Canada (DRH, EMS,YZ, TJ); NASA (FM, MDS, WHF, JAS); European Re- search Councilproject AQUMET, FET Proactive project QUIC, Spanish MINECO projectMAGO (Ref. FIS2011- 23520) and EPEC (FIS2014-62181-EXP), Catalan2014- SGR-1295, the European Regional Development Fund (FEDER) grantTEC2013-46168-R, and Fundacio Privada CELLEX (MWM, CA, WA, VP);New Brunswick Innovation Foundation (DRH). Part of the research was car-ried out at the Jet Propulsion Laboratory, California Institute of Technology,under a contract with the National Aeronautics and Space Administration.This work includes contributions of the National Institute of Standards andTechnology, which are not subject to U.S. copyright.

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